Assuming we know the local volatility function σ(S,t) for all S and t, how can we recover the corresponding implied volatility for a given strike K (and the other necessary parameters, S, r, T, t...)? What is the formal mathematical way to express the dependency σimplied(K) to the σ(S,t)s?

Can you also recommend documents where I can find numerical methodology?

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    $\begingroup$ Look for approximations based on the "most likely path" (Gatheral, Reghai etc.) $\endgroup$ – Quantuple Jan 18 '17 at 7:20
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    $\begingroup$ Alternatively, you could explicitly value the corresponding European plain vanilla options. E.g. use finite differences to solve the local volatility pricing PDE and then compute their implied volatilities. $\endgroup$ – LocalVolatility Jan 18 '17 at 15:25

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